Significant Figures And The Density Of Water - Version 1.5

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1 Significant Figures And The Density Of Water - Version 1.5 Michael J. Vitarelli Jr. Department of Chemistry and Chemical Biology Rutgers University, 610 Taylor Road, Piscataway, NJ I. INTRODUCTION Welcome to Introduction to Experimentation, 01:160:171. In this introductory laboratory the student will become acquainted with a few pieces of experimental equipment including beakers, burettes, ring-stands, and analytical balances. During the experiment students will use these items to determine the density of water. Along with learning how to use these items the student will become familiar with significant figures. This experiment does not have a pre-lab assignment nor a chemical hazards awareness form. However, all future experiments will. II. READING ANALYTICAL BALANCES AND BURETTES Burettes used in this laboratory are graduated to 0.1 ml; this is, there exists a thin dark line at every 0.1 ml increment. Even though there is no graduation for every hundredths of ml we will record volumes to the hundredths of ml. As an example, ml would be an acceptable measurement. The last decimal place is significant even though it is an approximation; significant figures discussed below. A further complication of this measurement is the meniscus. The meniscus is a curved surface of liquid caused by the attraction of the liquid to the burette vs. the attraction of the liquid to itself. While performing the measurement, keep the meniscus at eye level. This will avoid errors known as parallax errors. During each measurement record the volume measurement at the bottom of the meniscus. As mentioned the hundredths place is an approximation. Record this value to two hundredths of a ml, this corresponds approximately to the thickness of the dark lined graduation. Reading analytical balances are slightly easier. While using an analytical balance, measurements of mass should be recorded to the highest precision available. In this laboratory record the measurement of mass to the nearest mg (0.001 g). III. SIGNIFICANT FIGURES Why are significant figures important and why does one use them? Consider an example of two students: student one measures a mass and records the value as m 1 =2.1 grams. Student two measures a different mass and records the value as m 2 =4.67 grams. Now these students need to add these two masses together, m 1 +m 2, and record the combined mass, m 3. The question now arises, what is this value? Mathematically we add and obtain Is 6.77 grams the correct value for the mass of the combined system? Or is it some other value; possibly rounding 6.77 to 6.8? The answer turns out to be m 3 =6.8 grams, but why? Notice for m 1 =2.1 grams the mass is recorded to one decimal place. However, did the student round that number, or possibly the precision of the balance used is only to one decimal place? We are, of course, uncertain of the value of the second decimal place. The mass m 1 could have been any value between 2.05 and 2.14; both of these numbers would round to 2.1. Even though the mass m 2 is recorded with a precision of two decimal places, the mass

2 m 1 is recorded with a precision of only one decimal place. Thus the mass m 3 can only be recorded with a precision of one decimal place. In short, a calculated value can not be presented with more precision then the measured values it was calculated from. 2 IV. SIGNIFICANT FIGURES: RULES 1. Any non-zero digit is significant. Example: has 4 significant figures. 2. Zeros located between non-zeros are significant. Example: 2303 has 4 significant figures. 3. Zeros to the left of the last nonzero digit are NOT significant. Example: has only 2 significant figures. All the zeros in this number are not significant. 4. If a number contains a decimal point, zeros to the right of the last non-zero digit ARE significant. Example: has 6 significant figures. All the zeros in this number are significant. 5. If a number does NOT contain a decimal point then the numbers to the right of the last nonzero digit are ambiguous. Example, the number 200 may have 1, 2 or even 3 significant figures. We do not know if those zeros are significant. In this case the number has to be expressed in scientific notation to reveal the number of significant figures. 6. Significant figures only apply to measured quantities. Exact quantities or exact definitions of one unit in terms of another have an infinite number of significant figures. Examples: I have four oranges. The four would have an infinite number of significant figures. There are 60 minutes in one hour. This conversion would have an infinite number of significant figures. One final example would be the radius of a circle is half that of the diameter. Here the half has an infinite number of significant figures. 7. When adding or subtracting the result should be presented with the same number of digits after the decimal point as the measured value with the fewest. That is if you add and 3.1, the result will be 5.1 to one digit after the decimal point. 8. When multiplying or dividing the result should be presented with the same number of significant figures as the measured value with the fewest. That is if you multiply 2.00 and 4.0, the result will be 8.0 with two significant figures. V. PROCEDURE Water is a relatively safe and plentiful compound, thus in this first experiment the student will be using a beaker, burette, and an analytical balance to determine the density of water. Each student will perform this experiment on their own and perform the related calculations on their own. 1. Record the mass of an empty 50 ml beaker. Record this value in Table I. The mass of the empty 50 ml beaker will be the same for each of the five trials.

3 2. Add between 1.0 and 1.5 ml of water to your 50 ml beaker. Record this value in Table I. Do not discard this water. During each trial the student will be adding water to the 50 ml beaker. The volume of water in the beaker will always be increasing. Also, record the final and initial burette values in Table I. 3. Subtract the final and initial burette readings to obtain the volume of water added to the beaker. Record this value in Table I. 4. Using an analytical balance, measure the mass of the beaker with water. Record this value in Table I. From this value subtract the original value for the mass of the beaker. This will reveal the mass of the water in the beaker. Record this value in Table I. 5. Repeat this procedure four more times, adding water to the beaker each time. Never discard the water in the beaker, that is, until you have finished all five trials. 6. Plot the mass of water in grams vs. volume in water in ml on the first graph. Fit a straight line to this data. The slope of this line is the density of water which should be recorded in g/ml. Do not try to force the fit to go through the origin. In many cases experimental data certainly does not pass through the origin. 3 TABLE I: Experimental Data Measurement Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Mass of empty beaker [g] Mass of beaker + liquid [g] Mass of liquid [g] Initial burette reading [ml] Final Burette reading [ml] Volume of liquid [ml]

4 4 VI. GRAPH OF EXPERIMENTAL DATA Record slope with appropriate units: This is the density of water at room temperature. As mentioned, do not force the fit to pass through the origin. In an ideal experiment this would certainly occur here since zero volume would correspond to zero mass. However, no experiment is perfect or even ideal. Include title and appropriate axis labels to the graph.

5 5 VII. POST-LAB QUESTIONS 1. How many significant figures does the number have? 2. How many significant figures does the number have? 3. Perform the following calculation: , report the answer to the correct number of significant figures. 4. Perform the following calculation: 2.00 * 6.5, report the answer to the correct number of significant figures. 5. What is the answer to the following problem, reported to the correct number of significant figures? Try to complete this problem without a calculator. A similar problem may appear on the final exam (10.00) ( ) 6. Fit a straight line through the data that follows. Calculate the slope, x-intercept, and y- intercept. Present these values with the appropriate units. Record slope with appropriate units: Record x-intercept with appropriate units: Record y-intercept with appropriate units:

6 6 VIII. DATA TO FIT